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systems analysis and design using matlab

MATLAB An Introduction With Applications 6th Edition Amos Gilat - Solutions

35. The growth of a fish is often modeled by the von Bertalanffy growth model:where w is the weight and a and b are constants. Solve the equation for w for the case lb1/3, day–1, and lb. Make sure that the selected time span is just long enough so that the maximum weight is approached. What is
34. Use a MATLAB built-in function to numerically solve:for with Plot the solution.
33. Use a MATLAB built-in function to numerically solve:for with In one figure plot the numerical solution as a solid line and the exact solution as discrete points (10 equally spaced points).Exact solution: .
32. Use a MATLAB built-in function to numerically solve:for with In one figure plot the numerical solution as a solid line and the exact solution as discrete points.Exact solution: .
31. Use a MATLAB built-in function to numerically solve:for with Plot the numerical solution.
30. The Fresnel integrals are:and Calculate and for (use spacing of 0.05). In one figure plot two graphs—one of versus x and the other of versus x. In a second figure plot versus .
29. The orbit of Mercury is elliptical in shape, with km and km. The perimeter of an ellipse can be calculated by where . Determine the distance Mercury travels in one orbit. Calculate the average speed at which Mercury travels(in km/s) if one orbit takes about 88 days.
28. A cross-sectional area has the geometry of half an ellipse, as shown in the figure to the right. The coordinate of the centroid of the area can be calculated by:where A is the area given by , and is the moment of the area about the y axis, given by:Determine when mm and mm.
27. To estimate the surface area and volume of a football, the diameter of the ball is measured at different points along the ball. The surface area, S, and volume, V, can be determined by:and Use the data given in the table to determine the volume and surface area of the ball.
26. An approximate map of Lake Erie is shown in the figure. Use numerical integration to estimate the area of the lake. Make a list of the width of the lake as a function of x. Start with mi and use increments of 20 mi, such that the last point is . Compare the result with the actual area Lake
25. The variation of gravitational acceleration g with altitude y is given by:where km is the radius of the Earth, and m/s2 is the gravitational acceleration at sea level. The change in the gravitational poten-tial energy, U, of an object that is raised from the Earth is given by:Determine the
24. The length of a curve given by a parametric equation , is given by:The cardioid curve shown in the figure is given by:and with . Plot the cardioid with and determine the length of the curve.
23. The flow rate Q (volume of fluid per second)in a round pipe can be calculated by:For turbulent flow the velocity profile can be estimated by: . Determine Q for in.,, in./s.
22. The electric wire that connects the house to the pole has the shape of a catenary:By using the equation:determine the length of the wire.
21. Use numerical integration to approximate the size of the shaded area shown in the figure. Create a vector with values of x from 1 through 10 and estimate the corresponding y coordinate. Then, determine the area by using MATLAB’s builtin function trapz.
20. A rubber band is stretched by fixing one end pulling the other end. Measurements of the applied force at different displacements are given in the following table:Determine the work done by the force while stretching the rubber band.
19. The speed of a race car during the first 7 s of a race is given by:Determine the distance the car traveled during the first 7 s.
18. Use MATLAB to calculate the following integrals:(a) (b)
17. Use MATLAB to calculate the following integrals:(a) (b)
16. A 108-in.-long beam AB is attached to the wall with a pin at point A and to a 68-in.-long cable CD. A load lb is attached to the beam at point B. The tension in the cable T is given by:where L and LC are the lengths of the beam and the cable, respectively, and d is the distance from point A to
15. An RLC circuit with an alternating voltage source is shown. The source voltage is given by , where , in which is the driving frequency. The amplitude of the current, I, in this circuit is given by:where R and C are the resistance of the resistor and capacitance of the capacitor, respectively.
14. A prismatic box with equilateral triangular base is made from a equilateral triangular sheet with sides s by cutting off the corners and folding the edges along the dashed lines. For in., use MATLAB’s built-in function fminbnd to determine the value of x such that the box will have the
13. Using MATLAB’s built-in function fminbnd, determine the dimensions (radius r and height h)and the volume of the cylinder with the largest volume that can be made inside of a cone with a radius R of 20 in. and height H of 50 in.
12. A flat rectangular sheet of metal that is 70 in. wide and 120 in. long is formed to make a container with the geometry shown in the figure. (Additional flat metal pieces are attached at the ends.)Using MATLAB’s built-in function fminbnd, determine the value of h such that the container will
11. Using MATLAB’s built-in function fminbnd, determine the minimum and the maximum of the function
10. For fluid flow in a pipe, the Colebrook–White (or Colebrook) equation gives a relationship between the friction coefficient,f, and the Reynolds number:where k/d is the pipe relative roughness. Determine f if , and.
9. A series RLC circuit with an AC voltage source is shown. The amplitude of the current, I, in this circuit is given by:where in which is the driving frequency; R and C are the resistance of the resistor and capacitance of the capacitor, respectively; and is the amplitude of V. For the circuit in
8. An estimate of the minimum velocity required for a round flat stone to skip when it hits the water is given by (Lyderic Bocquet, “The Physics of Stone Skipping,” Am. J. Phys., vol. 71, no. 2, February 2003):where M and d are the stone mass and diameter, is the water density, C is a
7. The van der Waals equation gives a relationship between the pressure p(atm), volume V (L), and temperature T (K) for a real gas where n is the number of moles, (L atm)/(mol K) is the gas constant, and a (L2 atm/mol2) and b (L/mol) are material constants.Determine the volume of 1.5 mol of
6. The position s of the slider as a function of in the crank-slider mechanism shown is given by:Given in., in., and in., determine the angle , when in. (There are two solutions.)
5. The area A of a circle segment is given by:Determine the angle (in degrees) if in. and in2.
4. Determine the positive roots of the equation .
3. Determine the three roots of the equation .
2. Determine the solution of the equation .
1. Determine the two solutions of the equation between and .
35. The transmission of light through a transparent solid can be described by the equation:where is the transmitted intensity, is the intensity of the incident beam, is the absorption coefficient, L is the length of the transparent solid, and R is the fraction of light which is reflected at the
34. When rubber is stretched, its elongation is initially proportional to the applied force, but as it reaches about twice its original length, the force required to stretch the rubber increases rapidly. The force, as a function of elongation, that was required to stretch a rubber specimen that was
33. Curve-fit the data from the previous problem with a third-order polynomial.Use the polynomial to estimate y at . Make a plot of the points and the polynomial.
32. The relationship between two variables y and x is known to be:The following data points are given:Determine the constants a and b by curve-fitting the equation to the data points. Make a plot of y versus x. In the plot show the data points with markers and the curve-fitted equation with a solid
31. Use the data from Problem 30 for the following:(a) Fit the data with linear interpolation. Estimate the concentration at. Make a plot that shows the data points and curve made of interpolated points.(b) Fit the data with spline interpolation. Estimate the concentration at h. Make a plot that
30.Measurements of the concentration, C, of a substance during a chemical reaction at different times t are shown in the table.(a) Suppose that the data can be modeled with an equation in the form:Determine the coefficients a0, a1, and a2 such that the equation best fits the data. Use the equation
29. Estimated values of thermal conductivity of silicon at different temperatures are given in the following table.(a) Make a plot of k versus T using log scale on both axes.(b) Curve-fit the data with a second-order polynomial in which and . Once the coefficientsa, b, and c are determined, write
28. Write a user-defined function that determines the best fit of an exponential function of the form . Name the function [b,m] =expofit(x,y), where the input arguments x and y are vectors with the coordinates of the data points, and the output arguments b and m are the constants of the fitted
27. The standard air density, D (average of measurements made), at different heights, h, from sea level up to a height of 33 km is given below.(a) Make the following four plots of the data points (density as a function of height): (1) both axes with linear scale; (2) h with log axis, D with linear
26. The following points are given:(a) Fit the data with a first-order polynomial. Make a plot of the points and the polynomial.(b) Fit the data with a second-order polynomial. Make a plot of the points and the polynomial.(c) Fit the data with a third-order polynomial. Make a plot of the points and
25. Use the growth data from Problem 24 for the following:(a) Curve-fit the data with a third-order polynomial. Use the polynomial to estimate the height in day 40.(b) Fit the data with linear and spline interpolations and use each interpolation to estimate the height in day 40.In each part make a
24. Growth data of a sunflower plant is given in the following table:x –5 –4 –1 1 4 6 9 10 y 12 10 6 2 –3 –6 –11 –12 h (ft) –1,000 0 3,000 8,000 15,000 22,000 28,000 T ( °F ) 213.9 212 206.2 196.2 184.4 172.6 163.1 Year 1815 1845 1875 1905 1935 1965 Population(millions)8.3 19.7 The
23. The number of bacteria measured at different times t is given in the following table. Determine an exponential function in the form that best fits the data. Use the equation to estimate the number of bacteria after 5 h. Make a plot of the points and the equation.
22. The U.S. population in selected years between 1815 and 1965 is listed in the table below. Determine a quadratic equation in the form, where t is the number of years after 1800 and P is the population in millions, that best fits the data. Use the equation to estimate the population in 1915 (the
21. The boiling temperature of water at various altitudes h is given in the following table. Determine a linear equation in the form that best fits the data. Use the equation for calculating the boiling temperature at 5,000 m. Make a plot of the points and the equation.
20. The following data is given:(a) Use linear least-squares regression to determine the coefficients m and b in the function that best fits the data.(b) Make a plot that shows the function and the data points.
19. Consider the parabola:, and the point.(a) Write a polynomial expression for the distance d from point P to an arbitrary point Q on the parabola.(b) Make a plot of d versus y for.(c) Determine the coordinates of Q if (there are two points).(d) Determine the coordinates of Q that correspond to
18. A cylinder with base radius r and height h is constructed inside a sphere such that it is in contact with the surface of a sphere, as shown in the figure.The radius of the sphere is in.(a) Create a polynomial expression for the volume V of the cylinder in terms of h.(b) Make a plot of V versus
17. Write a user-defined function that calculates the maximum (or minimum) of a quadratic equation of the form:Name the function [x,y,w] = maxormin(a,b,c). The input arguments are the coefficientsa, b, andc. The output arguments are x, the coordinate of the maximum (or minimum); y, the maximum (or
16. Write a user-defined function that multiplies two polynomials. Name the function p=polymult(p1,p2). The two input arguments p1 and p2 are vectors of the coefficients of the two polynomials. The output argument p is the resulting polynomial.Use the function to multiply the following
15. Write a user-defined function that adds or subtracts two polynomials of any order. Name the function p=polyadd(p1,p2,operation). The first two input arguments p1 and p2 are the vectors of the coefficients of the two polynomials. (If the two polynomials are not of the same order, the function
14. The probability P of selecting three distinct numbers out of n numbers is calculated by:Determine how many numbers, n, should be in a lottery game such that the probability of matching three numbers out of n numbers will be at least 1/100,000, but not greater than 1/95,000.
13. A rectangular box (no top) is welded together using sheet metal. The length of the box’s base is 18 in. longer than its width. The total surface area of the sheet metal that is used is 2,500 in.2.(a) Using polynomials write an expression for the volume V in terms of x.(b) Make a plot of V
12. An aluminum container has the geometry shown in the figure (the bottom part is a rectangular box and the top is half a cylinder). The outside dimensions are shown. The wall thickness of the bottom and all the vertical walls is 2t, and the walls thickness of the cylindrical section is t.
11. A rectangular steel container (no top) has the outside dimensions shown in the figure. The thickness of the bottom surface is t, and the thickness of side walls is . Determine t if the weight of the container is 1,300 lb. The specific weight of steel is 0.284 lb/in.3.
10. The product of three distinct integers is 6,240. The sum of the numbers is 85. The difference between the largest and the smallest is 57. Using MATLAB’s built-in functions for operations with polynomials, determine the three integers.
9. The product of three integers with spacing of 3 between them (e.g., 9, 12, 15)is 11,960. Using MATLAB’s built-in functions for operations with polynomials, determine the three integers.
8. The product of four consecutive even integers is 1,488,384. Using MATLAB’s built-in function for operations with polynomials, determine the two integers.
7. Use MATLAB to divide the polynomial by the polynomial.
6. Use MATLAB to divide the polynomial by the polynomial .
5. Use MATLAB to carry out the following multiplication of polynomials:Plot the polynomial in the domain .
4. Use MATLAB to carry out the following multiplication of two polynomials:
3. Determine the polynomial that has roots at , ,, and . Make a plot of the polynomial in the domain.
2. Plot the polynomial in the domain .First create a vector for x, next use the polyval function to calculate y, and then use the plot function.
1. Plot the polynomial in the domain. First create a vector for x, next use the polyval function to calculate y, and then use the plot function.
40. In lottery the player has to guess correctly r numbers that are drawn out of n numbers. The probability, P, of guessing m numbers out of the r numbers can be calculated by the expression:where . Write a user-defined MATLAB function that calculates P. For the function name and arguments, use P =
39. The first derivative of a function at a point can be approximated with the four-point central difference formula:where h is a small number relative to . Write a user-defined function function (see Section 7.9) that calculates the derivative of a math function by using the four-point central
38. A circuit that filters out a certain frequency is shown in the figure. In this filter, the ratio of the magnitudes of the voltages is given by:where , and f is the frequency of the input signal.Write a user-defined MATLAB function that calculates the ratio of magnitudes. For the function name
37. The simple RC high-pass filter shown in the figure passes signals with frequencies higher than a certain cutoff frequency. The ratio of the magnitudes of the voltages is given by:where , and f is the frequency of the input signal.Write a user-defined MATLAB function that calculates the ratio of
36. The area moment of inertia of a rectangle about the axis passing through its centroid is . The moment of inertia about an axis x that is parallel to is given by , where A is the area of the rectangle, and is the distance between the two axes.Write a MATLAB user-defined function that determines
35. Write a user-defined function that determines the coordinate of the centroid of the Ishaped cross-sectional area shown in the figure.For the function name and arguments, use yc =centroidI(w,h,d,t), where the input arguments w, h,d, and t are the dimensions shown in the figure and the output
34. The Taylor series expansion for about is given by:where x is in radians. Write a user-defined function that determines using Taylor’s series expansion. For function name and arguments, use y=sinTay(x), where the input argument x is the angle in degrees and the output argument y is the value
33. In a lottery the player has to select several numbers out of a list. Write a user-defined function that generates a list of n integers that are uniformly distributed between the numbers a andb. All the selected numbers on the list must be different. For function name and arguments, use
32. Delta rosette is a set of three strain gages oriented at 120° relative to each other. The strain measured with each of the strain gages is , , and . The principal strains and can be calculated from the strains measured with the rosette by:Write a user-defined MATLAB function that determines
31. The shortest distance between two points on the surface of the globe (greatcircle distance) can be calculated by using the haversine formula. If and are the latitude and longitude of point 1 and and are the latitude and longitude of point 2, the great circle distance between the points is given
30. Write a user-defined MATLAB function that calculates the determinant of a matrix by using the formula:For the function name and arguments, use d3 = det3by3(A), where the input argument A is the matrix and the output argument d3 is the value of the determinant. Write the code of det3by3 such
29. Write a user-defined MATLAB function that finds the largest element of a matrix. For the function name and arguments, use [Em,rc] = matrixmax(A), where A is any size matrix. The output argument Em is the value of the largest element, and rc is a two-element vector with the address of the
28. Write a user-defined function that sorts the elements of a matrix. For the function name and arguments, use B = matrixsort(A), where A is any size matrix and B is a matrix of the same size with the elements of A rearranged in descending order column after column with the (1,1) element the
27. Write a user-defined function that sorts the elements of a vector from the largest to the smallest. For the function name and arguments, use y=downsort(x). The input to the function is a vector x of any length, and the output y is a vector in which the elements of x are arranged in a descending
26. Write a user-defined function that determines the value that occurs most often in a set of data that is given in a two-dimensional matrix. For the function name and arguments, use [v, q] =matrixmode(x). The input argument x is a matrix of any size with numerical values, and the output arguments
25. Write a user-defined function that determines the polar coordinates of a point from the Cartesian coordinates in a two-dimensional plane. For the function name and arguments, use [th rad]=CartToPolar(x,y).The input arguments are the x and y coordinates of the point, and the output arguments are
24. The harmonic mean H of a set of n positive numbers is defined by:Write a user-defined function that calculates the harmonic mean of a set of numbers. For function name and arguments use G=Harmean(x), where the input argux ment x is a vector of numbers (any length) and the output argument H is
23. Write a user-defined function that determines if a number is a prime number.Name the function pr=Trueprime(m), where the input arguments m is a positive integer and the output argument pr is 1 if m is a prime number and 0 if m is not a prime number. Do not use MATLAB’s built-in functions
22. In polar coordinates a two-dimensional vector is given by its radius and angle . Write a user-defined MATLAB function that adds two vectors that are given in polar coordinates. For the function name and arguments, use[r th]= AddVecPol(r1,th1,r2,th2), where the input arguments are and, and the
21. Write a user-defined function that plots an ellipse with axes that are parallel to the x and y axes, given the coordinates of its vertices and the coordinates of another point that the ellipse passes through. For the function name and arguments, use ellipseplot(A,B,C). The input arguments A and
20. Write a user-defined function that plots a triangle and the circle that is inscribed inside, given the coordinates of its vertices. For the function name and arguments, use TriCirc(A,B,C). The input arguments are vectors with the x and y coordinates of the vertices, respectively. This function
19. Write a user-defined MATLAB function that converts integers written in decimal form to binary form. Name the function b=Bina(d), where the input argument d is the integer to be converted and the output argument b is a vector with 1s and 0s that represents the number in binary form. The largest
18. Write a user-defined function that determines the location of the center and the radius of a circle that passes through three given points in a plane. The function also creates a plot that shows the circle and the points. For the function name and arguments, use [C R]=Circle3Pts(A,B,C). The
17. As shown in the figure, the area of a convex polygon can be calculated by adding the area of the triangles that the polygon can be divided into. Write a userdefined MATLAB function that calculates the area of a convex n-sided polygon. For the function name and arguments, use A = APolygon(Crd).
16. The area of a triangle ABC can be calculated by:where AB is the vector from vertex A to vertex B and AC is the vector from vertex A to vertex C. Write a user-defined MATLAB function that determines the area of a triangle given its vertices’ coordinates. For the function name and arguments,
15. Write a user-defined MATLAB function that determines the cross product of two vectors. For the function name and arguments, use w=crosspro(u,v). The input arguments to the function are the two vectors, which can be two- or three-dimensional. The output w is the result (a vector). Use the
14. Write a user-defined MATLAB function that determines the unit vector in the direction of the line that connects two points (A and B) in space. For the function name and arguments, use n = unitvec(A,B). The input to the function are two vectors A and B, each with the Cartesian coordinates of the
13 Write a user-defined MATLAB function that determines the time elapsed between two events during a day. For the function name and arguments, use dt = timediff(TA,ap1,TB,ap2). The input arguments to the function are:TA is a two-element vector with the time of the first event. The first element is
12. Write a user-defined MATLAB function that determines the angle that forms by the intersection of two lines. For the function name and arguments, use th=anglines(A,B,C). The input arguments to the function are vectors with the coordinates of the points A, B, and C, as shown in the figure, which
11. Write a user-defined function that calculates grade point average (GPA) on a scale of 0 to 5, where , , , , and . For the function name and arguments, use GPA = GradePtAve(G,C). The input argument G is a vector whose elements are letter grades A, B, C, D, or F entered as a string (e.g.,

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